Oct 08, 2015
Qualitative Spatial Reasoning overLine-Region Relations
Leena and Sibel
Knowledge RepresentationSeminar Presentation
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Agenda
Motivation
9-Intersection
Snapshot Model
Smooth-Transition Model
Evaluation
Summary
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Motivation
I Modeling spatial relations
I How do humans conceptualize spatial relations?
I Strong correlation between Perceptual space and LanguageSpace
I Understanding cognitive perceptual groupings
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Agenda
Motivation
9-Intersection
Snapshot Model
Smooth-Transition Model
Evaluation
Summary
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9-Intersection
GoalA computational model to describe conceptual neighborhoods andenable the definition of a similarity metric for line region relations.
Conceptual Similarity: Which pairs of relationships aresimilar?
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Formal Definitions
LineA sequence of 1...n connected 1-cells between two geometricallyindependent 0-cells such that they neither cross each other norform cycles.
I Interior, Boundary, Exterior
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Formal Definitions(contd.,)
Region
A region is defined as a connected, homogeneously 2-dimensional2-cell. Its boundary forms a Jordan curve separating the regionsexterior from its interior.
I Interior, Boundary, Exterior
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Adjacency
Topological adjacency
I Adjacent(Interior A0) = A
I Adjacent(Boundary A) = A0andA
I Adjacent(Exterior A) = A
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Adjacency
Topological adjacency
I Adjacent(Interior A0) = A
I Adjacent(Boundary A) = A0andA
I Adjacent(Exterior A) = A
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Adjacency
Topological adjacency
I Adjacent(Interior A0) = A
I Adjacent(Boundary A) = A0andA
I Adjacent(Exterior A) = A
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Adjacency
Topological adjacency
I Adjacent(Interior A0) = A
I Adjacent(Boundary A) = A0andA
I Adjacent(Exterior A) = A
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9-Intersection(contd..,)
Topological adjacency
I 9 intersections between the different topological parts of a lineand a region
The 9-intersection Matrix(M) L0 R0 L0 R L0 RL R0 L R L RL R0 L R L R
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9-Intersection(contd..,)
I Binary assignment to intersections(,)I 512 possible instances of M
I 19 of 512 instances can actually be realized.
Example
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9-Intersection(contd..,)
I Binary assignment to intersections(,)I 512 possible instances of M
I 19 of 512 instances can actually be realized.
Example
Compute the values of the matrix... 14/41
Geometric interpretations
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Agenda
Motivation
9-Intersection
Snapshot Model
Smooth-Transition Model
Evaluation
Summary
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Snapshot Model
I A model of conceptual neighborhood among topologicalrelations between a line and a region.
Characteristics
I No prior knowledge of the potential transformations thatcould lead from one configuration to the other.
I Comparison on the basis of a pre-defined distance metric
Differences of Intersections(= 0, = 1)
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Defining topological neighbors
Distance between any two relationships rA, rB is given by:
TrA,rB =i=0
j=0
|MA[i , j ]MB [i , j ]| (1)
I It is the count of differences of empty/non empty entries ofcorresponding elements in the 9 intersections.
I Shortest Nonzero distance between relations is 1
I Spatial relations with the shortest non zero distance areconsidered topological neighbors.
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Example 1
Are these relations neighbors according to the snapshotmodel?
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Example 2
Another example... Are these relations topological neighbors?
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Conceptual neighborhoods derived from thesnapshot model
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Agenda
Motivation
9-Intersection
Snapshot Model
Smooth-Transition Model
Evaluation
Summary
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Smooth-Transition Model
Smooth TransitionAn infinitesimally small deformation that changes the topologicalrelation between the line and the region
Examples and Counterexamples
conceptual neighborsconceptual neighbors
not conceptualneighbors
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Formalization
A smooth transition occurs by moving around the lines
1. boundary nodes
Q: Do they intersect with the same region part?Transition Rule 1 if YesTransition Rule 2 if No
2. interior
Transition Rule 3 to extend the intersection area andTransition Rule 4 to reduce it
What this means for the 9-intersection:An entry or its adjacent entries gets changed from to or v.v.
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One More Thing...
Definition (Extent of a line part i)
I Denoted by #M[i , ]
I Count of intersections betw. line part i and the region parts
I #M[i , ] in the interval [0 . . . 3]
Examples
I extent of the lines boundary is either 1 (if both nodes arelocated in the same region part) or 2 (if the nodes are locatedin different parts of the region)
I extent of a lines exterior is always 3
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Transition Rule 1
If the lines two boundaries intersect with the same region part,then extend the intersection to either of the adjacent region parts:
#M[, ] = 1 = i(M[, i ] = ) : MN [, adjacent(i)] :=
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Transition Rule 2
If the lines two boundaries intersect with two different regionparts then move either intersection to the adjacent region part:
#M[, ] = 2 = i(M[, i ] = ) :MN [, i ] := MN [, adjacent(i)] :=
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Transition Rule 3
Extend the lines interior-intersection to either of the adjacentregion parts:
i(M[, i ] = ) : MN [, adjacent(i)] :=
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Transition Rule 4
Reduce the lines interior intersection on either of the adjacentregion parts.
#M[, ] = 2 = i(M[, i ] = ) : MN [, i ] := #M[, ] = 3 = i(i 6= ) : MN [, i ] :=
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Additional Consistency Constraints
1. If the lines interior intersects with the regions interior andexterior, then the lines interior must also intersect with theregions boundary.
M[, ] = M[, ] = = M[, ] :=
2. If the lines boundary intersects with the regions interior(exterior) then the lines interior must intersect with theregions interior (exterior) as well.
M[, ] = = M[, ] := M[, ] = = M[, ] :=
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Resulting Neighborhood Graph
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Comparison
Snapshot Model Smooth-Transition Model
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Agenda
Motivation
9-Intersection
Snapshot Model
Smooth-Transition Model
Evaluation
Summary
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Setup
I 2 geometrically distinct placements of the line for each of the19 topologically distinct relations
I a total of 38 diagrams each showing a line and a region
I line road, region parkI parks in all diagrams same size and shape
I 28 participants
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Setup (cont.)
Q: Find the pair that is topologically identical from among allgeometrically distinct diagrams.
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Setup (cont.)
Q: Find the pair that is topologically identical from among allgeometrically distinct diagrams.
A: The right and middle examples in the lower row.
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Task
I arrange the sketches into several groups, such that you woulduse the same verbal description for the spatial relationshipbetween the road and the park for every sketch in each group
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Goal
Goal
I analyse how the subjects formed groups of similar relations
I check similarity with presented conceptual neighborhoodmodels
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Results
Each spatial relation could be grouped as many as 112 times (4pairs times 28 subjects) with each other relation.
Number of times conceptual neighbors are grouped:
. . . 1 min max mean %2
snapshot model only 2 10 16 13.0 11.6smooth-transition model only 12 0 66 17.3 15.4both models 26 0 78 33.6 29.5neither model 131 - - 6.0 5.3
1Number of relations that are conceptual neighbors2percentage = mean / 112
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Agenda
Motivation
9-Intersection
Snapshot Model
Smooth-Transition Model
Evaluation
Summary
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Summary
Two Conceptual Neighborhood Models:
1. Snapshot Model
2. Smooth-Transition Model
Finding: Almost identical Conceptual-Neighborhood Graphs
Findings from the Human-Subject Experiment:
I models correspond largely to the way humans conceptualizesimilarity about line-region relations
I smooth-transition model captures more aspects of thesimilarity of topological line-region relations than the snapshotmodel
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References I
I Egenhofer, Max J., and David M. Mark. Modellingconceptual neighbourhoods of topological line-regionrelations. International journal of geographical informationsystems 9.5 (1995): 555-565.
I Mark, David M., and Max J. Egenhofer. Modeling spatialrelations between lines and regions: combining formalmathematical models and human subjects testing.Cartography and geographic information systems 21.4 (1994):195-212.
I Egenhofer, Max J., and A. Rashid Shariff. Metric details fornatural-language spatial relations. ACM Transactions onInformation Systems (TOIS) 16.4 (1998): 295-321.
I Talmy, Leonard. How language structures space. Springer US,1983.
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References II
I Clark, Herbert H. Space, time, semantics, and the child.(1973).
I Moore, Timothy E. Cognitive Development and theAcquisition of Language. (1973).
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Motivation9-IntersectionSnapshot ModelSmooth-Transition ModelEvaluationSummary